On a System of Difference Equations

Özkan, Ozan; Kurbanli, Abdullah Selçuk

doi:https://doi.org/10.1155/2013/970316

Abstract

We have investigated the periodical solutions of the system of rational difference equations , and where .

1. Introduction

Recently, a great interest has arisen on studying difference equation systems. One of the reasons for that is the necessity for some techniques which can be used in investigating equations which originate in mathematical models to describe real-life situations such as population biology, economics, probability theory, genetics, and psychology. There are many papers related to the difference equations system.

In [1], Kurbanli et al. studied the periodicity of solutions of the system of rational difference equations

In [2], Çinar studied the solutions of the systems of difference equations

In [3, 4], Özban studied the positive solutions of the system of rational difference equations

In [5–16], Elsayed studied a variety of systems of rational difference equations; for more, see references.

In this paper, we have investigated the periodical solutions of the system of difference equations where the initial conditions are arbitrary real numbers.

2. Main Results

Theorem 1. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system Also, assume that , , , and . Then, all six-period solutions of (5) are as follows:

Proof. For , we have For , assume that are true. Also, we have

Theorem 2. Let , , , , , , , , and be arbitrary real numbers, and let be a solution of the system Also, assume that , , , and . Then, all six-period solutions of (10) are as follows:

Proof. For , we have For , assume that are true. Also, we have

The following corollary follows from Theorem 1.

Corollary 3. The following conclusions are valid for :(i),(ii),(iii),(iv).

The following corollary follows from Theorem 2.

Corollary 4. The following conclusions are valid for :(i),(ii),(iii),(iv).

References

A. S. Kurbanli, C. Çinar, and D. Şımşek, “On the periodicity of solutions of the system of rational difference equations $x_{n + 1} = x_{n - 1} + y_{n} / y_{n} x_{n - 1} - 1$ , $y_{n + 1} = y_{n - 1} + x_{n} / x_{n} y_{n - 1} - 1$ ,” Applied Mathematics, vol. 2, no. 4, pp. 410–413, 2011. View at: Publisher Site | Google Scholar | MathSciNet
C. Çinar, “On the positive solutions of the difference equation system $x_{n + 1} = 1 / y_{n}$ , $y_{n + 1} = y_{n} / x_{n - 1} y_{n - 1}$ ,” Applied Mathematics and Computation, vol. 158, no. 2, pp. 303–305, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Y. Özban, “On the system of rational difference equations $x_{n} = a / y_{n - 3}$ , $y_{n} = b y_{n - 3} / x_{n - q} y_{n - q}$ ,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 833–837, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. Y. Özban, “On the positive solutions of the system of rational difference equations $x_{n + 1} = 1 / y_{n - k}$ , $y_{n + 1} = x_{n} / x_{n - m} y_{n - m - k}$ ,” Journal of Mathematical Analysis and Applications, vol. 323, no. 1, pp. 26–32, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “Solutions of rational difference systems of order two $x_{n + 1} = α + x_{n - m} / x^{k_{n}}$ ,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 378–384, 2012. View at: Publisher Site | Google Scholar | MathSciNet
E. M. Elsayed, “On the solutions of higher order rational system of recursive sequences,” Mathematica Balkanica. New Series, vol. 22, no. 3-4, pp. 287–296, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “Dynamics of a recursive sequence of higher order,” Communications on Applied Nonlinear Analysis, vol. 16, no. 2, pp. 37–50, 2009. View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “On the solutions of a rational system of difference equations,” Polytechnica Posnaniensis, no. 45, pp. 25–36, 2010. View at: Google Scholar | MathSciNet
E. M. Elsayed, “On the solutions of a rational system of difference equations,” Polytechnica Posnaniensis, no. 45, pp. 25–36, 2010. View at: Google Scholar | MathSciNet
E. M. Elsayed, “Dynamics of recursive sequence of order two,” Kyungpook Mathematical Journal, vol. 50, no. 4, pp. 483–497, 2010. View at: Publisher Site | Google Scholar | MathSciNet
E. M. M. Elsayed, “Behavior of a rational recursive sequences,” Mathematica, vol. 56, no. 1, pp. 27–42, 2011. View at: Google Scholar | MathSciNet
E. M. Elsayed, “Solution of a recursive sequence of order ten,” General Mathematics, vol. 19, no. 1, pp. 145–162, 2011. View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “Solution and attractivity for a rational recursive sequence,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 982309, 17 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “On the solution of some difference equations,” European Journal of Pure and Applied Mathematics, vol. 4, no. 3, pp. 287–303, 2011. View at: Google Scholar | MathSciNet
E. M. Elsayed, M. M. El-Dessoky, and A. Alotaibi, “On the solutions of a general system of difference equations,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 892571, 12 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
E. M. Elsayed, “On the dynamics of a higher-order rational recursive sequence,” Communications in Mathematical Analysis, vol. 12, no. 1, pp. 117–133, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet

Copyright

Copyright © 2013 Ozan Özkan and Abdullah Selçuk Kurbanli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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